Relation and Function Concepts Explained

TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
Week 4
Relation and Function I

Relation and FunctionRelation and Function
Overview
Obviously, we do not realize that there many connections
are happened in our circumtances. For examples, day and
night happens because of earth rotation, all students in
math are also connected to other subjects and so on.
Strictly speaking, something happens because of other
Strictly speaking, something happens because of other
subject called “reason”.
Relations can be used to solve problems such as
determining which pairs of cities are linked by airline flights
in a network, finding a viable order for the different phases
of a complicated project, or producing a useful way to store
information in computer databases.
For couple weeks later, you all will be introduced this
“connection” in mathematic’s view. And we shall learn to
“map” or “transform” the “connection”.

Objectives
Cartesian Product
Relation
Invers Relation
Invers Relation
Pictoral Repesentation of Relation
Composition of Relation
Relation Properties

Cartesian Product
Consider two sets A and B. The set of all ordered
pairs (a, b) where a∈A and b∈B is called the
pairs (a, b) where a∈A and b∈B is called the
product, or Cartesian product, of A and B.
The short designation of this product is A x B,
which is read “A cross B”.

Ex.
Let A={1, 2} and B={a, b, c}.
Then
AxB {(1,a},(1,b),(1,c),(2,a),(2,b),(2,c)}
BxA {(a, 1), (a,2), (b, 1), (b,2), (c,1),(c,2)}
AxA {(1, 1), (1,2), (2,1), (2,2)}
From the example above we can conclude, that,
First,
First,
A x B ≠≠≠≠ B x A
The Cartesian product deals with ordered pairs, so naturally the order in
which the sets are considered is important.
Second, using n(s) for the number of elements in a set S, we have
n(A x B) = n(A) . n(B) = 2 x 3 = 6
Therefore, there will be 26 = 64 relation from A to B
So…..what is relation?????

Relation
Relation is just a subset of the Cartesian product
of the sets.
Definition.
Let A and B be sets. A binary relation or, simply,
Let A and B be sets. A binary relation or, simply,
relation from A to B is a subset of A x B.
In other words, a binary relation from A to B is a
set R of ordered pairs where the first element
(domain) of each ordered pair comes from A and
the second element (codomain or range) comes
from B.

We use the notation a R b to denote that (a, b) ∈ R
and a R b to denote that (a, b) ∉ R.
Moreover, when (a, b) belongs to R, a is said to be
related to b by R.
/
Assume C= {1,2,3} and D ={x,y,z} and let R {(1,y), (1,z),
(3,y)}. Put the R or R for the followings:
1…X 1…Y 1…Z
2…X 2…Y 2…Z
3…X 3…Y 3…Z
/

Invers Relation
The invers relation of set is defined as the opposite
mapping of relation itself.
Let R be any relation from a set A to a set B. The
inverse of R, denoted by R-1, is the relation from B
to A which consists of those ordered pairs which,
when reversed, belong to R; that is,
R-1= {(b,a): (a,b) ∈ R}

Ex.
Let R = {(1,y), (1,z), (3,y)} from A = {1,2,3} to
B = {x,y,z}, then
R-1 = {(y, 1), (z, 1), (y,3)}

Pictoral Repesentation of Relation
Arrow Diagram

Table Representation

Matrice Representation
Suppose R is the relation from A to B, where
A={ a1,a2,a3,…,am} and B={ b1,b2,b3,…,bn}.
The relation can be describe in matrice M=[mij] as
The relation can be describe in matrice M=[mij] as
folow:

Ex.
a1 = 2
a2 = 3
a3 = 4
a3 = 4
b1 = 2
b2 = 4
b3 = 8
b4 = 9
b5 = 15

Directed Graph
First we write down the elements of the set, and
then we drawn an arrow from each element x to
each element y whenever x is related to y.
each element y whenever x is related to y.
The point is, directed graph does not show the
relation between one set to the other. It just shows
the relation among the element inside the set.
Ex. R is relation on the set A = {1,2,3,4}
R = {(1,2), (2,2), (2,4), (3,2), (3,4), (4,1), (4,3)}

Prac.
Show the relation from
the directed graph
Bandung
Jakarta Surabaya
Medan
Makassar
Kupang

Composition of Relation
Suppose A, B, and C be sets, and let R be a
relation from A to B and let S be a relation
from B to C. R ⊆ A x B and S ⊆ B x C.
from B to C. R ⊆ A x B and S ⊆ B x C.
Then R and S give rise to a relation from A
to C, which is denoted by RoS and defined
as

Ex.
Assume A= {1,2,3,4}, B ={a,b,c,d}, C ={x,y,z}
and let R= {(1,a), (2,d), (3,a) (3,b), (3,d)} and
S ={(b,x), (b,z), (c,y), (d,z)} . Show the
relation a(RoS)c!
relation a(RoS)c!

From the picture we can observe that there is an arrow
from 2 to d which is followed by an arrow from d to z. We
can view these two arrows as a “path” which “connects” the
element 2 ∈ A to the element z ∈ C. Thus,
2(R o S)z since 2Rd and dSz
Similarly there is a path from 3 to x and a path from 3 to z.
Hence,
3(R o S)x and 3(R o S)z
No other element of A is connected to an element of C.
Therefore, the composition of relations R o S gives
RoS= {(2,z), (3,x), (3,z)}

Soal :
R = {(1, 2), (1, 6), (2, 4), (3, 4), (3, 6), (3, 8)}
S = {(2, u), (4, s), (4, t), (6, t), (8, u)}
Gambarkan grafiknya dan tentukan R o S
Gambarkan grafiknya dan tentukan R o S

R o S = {(1, u), (1, t), (2, s), (2, t), (3, s), (3, t), (3, u) }

Exercises :
1

2.

Relation and Function Concepts Explained

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